If a line in the space makes angle α, β and γ with the coordinate axes, then cos 2 α+cos 2 β+cos 2 γ+sin^2 α +sin^2 β +sin^2 γ equals

Correct answer: (C) 1. cos 2 α+cos 2 β+cos 2 γ+sin^2 α+sin^2 β +sin^2 γ = (cos^2 α-sin^2 α )+ (cos^2 β-sin^2 β ) + (cos^2 γ-sin^2 γ )+sin^2 α+sin^2 β+sin^2 γ =cos^2 α+cos^2 β+cos^2 γ =1

Q100 / 160 · AP EAMCET 2009 · Mathematics

If a line in the space makes angle $\alpha, \beta$ and $\gamma$ with the coordinate axes, then $\begin{aligned} \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma+\sin ^2 \alpha & +\sin ^2 \beta \\ & +\sin ^2 \gamma \text { equals } \end{aligned}$

A
-1
B
0
Correct answer
C
1
D
2

Solution

$\begin{aligned} & \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma+\sin ^2 \alpha+\sin ^2 \beta \\ & \quad+\sin ^2 \gamma \\ & =\left(\cos ^2 \alpha-\sin ^2 \alpha\right)+\left(\cos ^2 \beta-\sin ^2 \beta\right) \\ & \quad+\left(\cos ^2 \gamma-\sin ^2 \gamma\right)+\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma \\ & =\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma \\ & =1\end{aligned}$

Hence, the correct answer is option: 'C'.

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